On Linear Convergence of Gradient-Type Minimization Algorithms

M. Gaviano and D. Lera
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M. Gaviano: University of Cagliari
D. Lera: University of Cagliari

Journal of Optimization Theory and Applications, 1998, vol. 98, issue 2, No 11, 475-487

Abstract: Abstract Given the minimization problem of a real-valued function $$f\left( x \right),x \in \Re ^n $$ let A be any algorithm of type $$x_{i + 1} + \lambda _i h_i $$ with $$\lambda _i\in \Re ,h_i \in \Re ^\mathfrak{n} ,$$ $$ - h_i^T \nabla f\left( {x_i } \right) \geqslant \rho \left\| {h_i } \right\|\left\| {\nabla f\left( {x_i } \right)} \right\|,\rho\in \left( {{\text{0,1}}} \right) $$ that converges to a local minimum $$x^* \in f\left( x \right)$$ . In this note, new assumptions on f(x) under which A converges linearly to x* are established. These include the ones introduced in the literature which involve the uniform convexity of f(x).

Keywords: Convex programming; descent algorithms; convergence rates; optimization algorithms (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1023/A:1022601920647

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